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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>In this section, we only consider (Case a). Thus, we have two solutions:</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_5.html ./knowl/eq3_5_3.html">
\begin{equation*}
y_1(x)=e^{r_1 x},\quad y_2(x)=e^{r_2 x}.
\end{equation*}
</div>
<p class="continuation">It can be verified that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_5.html ./knowl/eq3_5_3.html">
\begin{equation}
y(x)=C_1 e^{r_1 x}+C_2 e^{r_2 x},\tag{3.1.8}
\end{equation}
</div>
<p class="continuation">satisfies the (<a href="" class="xref" data-knowl="./knowl/eq3_5.html" title="Equation 3.1.6">(3.1.6)</a>) and thus is also a solution. Here, <span class="process-math">\(C_1, C_2\)</span> are arbitrary constants. It turns out that (<a href="" class="xref" data-knowl="./knowl/eq3_5_3.html" title="Equation 3.1.8">(3.1.8)</a>) contains every possible solutions, i.e., it is the <dfn class="terminology">general solution</dfn> (this will be justified later on).</p>
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